Introduction to Elementary Logic
Content
Philosophy 211:
Elementary Logic
Emi Okayasu
Summer 2015
The Plan
• Introductions and Introduction to the course
‒Logistics
‒Some basics
• The Goal
• The tools we’ll use to reach the goal
The Syllabus
• Any questions?
Is this class curved?
Is this class curved?
Bad News
Good News
Is this class curved?
Bad News
• Even better news: No, this class is not curved.
Good News
The Plan
• Introductions and Introduction to the course
‒Logistics ✓
‒Some basics
• The Goal
• The tools we’ll use to reach the goal
Why study logic?
• What is logic anyway?
‒Logic is the study of correct reasoning
‒Logic is the beginning of wisdom, not the
end (Leonard Nimoy, aka Spock)
No, but really…what does a logician do?
• Logicians’ primary goal is to detect valid arguments.
• For example:
My dog is friendly.
So, your dog must be friendly too.
vs.
Either Yoshi or Mario won the race.
Yoshi didn’t win, so Mario won the race.
No, but really…what does a logician do?
• Logicians’ primary goal is to detect valid arguments.
• For example:
My dog is friendly.
So, your dog must be friendly too.
X
vs.
Either Yoshi or Mario won the race.
Yoshi didn’t win, so Mario won the race.
✓
Aristotle
• Aristotle (300s BC) was the first
person to systematically study and
categorize valid arguments.
• Aristotelian logic (more on this soon)
remained the gold standard of logic
for over 2000 years.
A note of caution
• Logic can be really tricky. In order to do logic well, we need to be
really precise.
• Specifically, we’ll be defining terms that you might already be
familiar with in new ways.
‒For example, saying that “Your point is valid” is perfectly
acceptable in normal conversation.
‒In this course, however, we will be defining validity a little
differently, so beware!
Arguments
• “An argument is a group of
statements, one or more of which
(the premises) are claimed to provide
support for, or reasons to believe,
one of the others (the conclusion)”
(from page 1 of your textbook)
Examples of Arguments
It’s not the case that both
Sherlock and Moriarty are alive.
Sherlock is not alive. So, Moriarty
is alive.
“Titanium combines readily with
oxygen, nitrogen, and hydrogen, all
of which have an adverse effect on
its mechanical properties. As a
result, titanium must be processed
in their absence.”
-From p. 14 of your book
Premise-Conclusion Form
• Most of the arguments that you’ll encounter in this class will be
presented like this:
It’s not the case that both
Sherlock and Moriarty are
alive.
Sherlock is alive.
Moriarty is not alive.
Titanium combines readily with
oxygen, nitrogen, and hydrogen.
Oxygen, nitrogen, and hydrogen all
have an adverse effect on titanium’s
mechanical properties.
Titanium must be processed in the
absence of oxygen, nitrogen, and
hydrogen.
Validity
• Validity is a property of arguments
‒ (Why a logician would never say you have a “valid point”!)
• An argument is valid when it is impossible for the conclusion to be
false if the premises are true.
• If it’s possible for the conclusion to be false even though all the
premises are true, we say the argument is invalid.
Applying Validity: test yourself
It’s not the case that both Sherlock and
Moriarty are alive.
Sherlock is alive.
Moriarty is not alive.
My dog is friendly.
Your dog is friendly.
Either Yoshi or Mario wins the race.
Yoshi doesn’t win.
Mario wins the race.
Titanium combines readily with
oxygen, nitrogen, and hydrogen.
Oxygen, nitrogen, and hydrogen all
have an adverse effect on titanium’s
mechanical properties.
Titanium must be processed in the
absence of oxygen, nitrogen, and
hydrogen.
The space of all possible arguments
All Arguments
Valid Arguments
The Weirdness of Validity (part I)
• It seems like our definition of validity (which is supposed to be a
good property of an argument) is broken.
• What’s the problem?
‒Validity means that the premises guarantee the truth of the
conclusion. In other words, if the premises are true, then the
conclusion is 100% certain to be true too.
For example…
The sun rose this morning.
The sun rose yesterday morning.
The sun rose the day before yesterday’s morning.
The sun will rise tomorrow morning.
Some arguments are not aiming to establish the truth of the
conclusion 100%. Some arguments are fine with establishing the
truth of the conclusion to 99% (or some other high percentage)
Inductive vs Deductive Arguments
Inductive Arguments
Deductive Arguments
Valid Arguments
Inductive vs Deductive Arguments
Inductive Arguments
Deductive Arguments
Valid Arguments
Deductive Arguments
• In this course, we are only going to focus what makes a good
deductive argument.
‒If you’re interested in what makes a good inductive argument,
take PHILOS 210
• Validity is a property that only applies to deductive arguments.
• You can assume (unless otherwise instructed) that whatever
argument you come across in this course is a deductive argument.
The Plan
• Introduction to the course
‒Logistics ✓
‒Some basics ✓
• The Goal
• The tools we’ll use to reach the goal
The Goal
• Now that we have some of the basics down, we can be more
specific about our goal:
• The goal of this course is to learn how to recognize what makes a
valid (or invalid) deductive argument, and then be able to prove
that a deductive argument is either valid or invalid.
Aristotle’s Observation
An
argument
is valid by
virtue of its
form.
-Aristotle
The form of an argument?
• Consider this argument again:
Either Mario or Yoshi wins
the race.
Yoshi does not win the race
Mario wins the race
The form of an argument?
• Consider this argument again:
Either Mario or Yoshi wins
the race.
Yoshi does not win the race
Mario wins the race
Either I eat museli or yogurt
for breakfast.
I do not eat yogurt for
breakfast.
I eat museli for breakfast.
The form of an argument
• Both these arguments have the same form:
‒ P1 says that there are two possibilities, and at least one of them must
happen.
‒ P2 says that one of the two options doesn’t happen.
‒ So, the conclusion says that the other option must have happened.
• Since both arguments have the same form, they must either both
be valid or both be invalid.
• One of the things we’ll be learning is how to translate an English
argument into a FORMal language (i.e. a language that captures
the form of the argument)
The form of an argument?
• We can represent the fact that these arguments have the same
form by writing them with symbols:
Either Mario or Yoshi wins the race.
Yoshi does not win the race
Mario wins the race
Either I eat museli or yogurt for breakfast.
I do not eat yogurt for breakfast.
I eat museli for breakfast.
MvY
~Y
M
Some Practice Problems
• Determine which of the following are valid. Do any have the same
form?
Pikachu is an electric pokemon.
Pikachu is a pokemon.
Superman can fly.
Superman is Clark Kent.
Clark Kent can fly.
Charizard is a pokemon.
Charizard is a fire pokemon.
All men are mortal.
Socrates is mortal.
Socrates is a man.
Valid vs Invalid Arguments
Deductive Arguments
Valid Arguments
The weirdness of Validity (part II)
• Hopefully you’re getting the hang of validity.
• Now consider this argument:
If there is green cheese on the moon, then Martians will harvest it.
There is green cheese on the moon.
Martians will harvest the green cheese on the moon
• Hopefully I don’t have to convince you that this argument is
completely crazy. BUT, it is valid (!?!)
Why?
Deductive Arguments
Valid Arguments
The weirdness of Validity (part II)
• According to our definition of validity, some pretty crazy
arguments are considered valid.
• But, we don’t want to call the green-cheese/Martian argument a
good argument.
• Validity does not guarantee that an argument is good.
• To help solve this problem, we’re going to introduce another
criteria for evaluating arguments: soundness
Soundness
• Soundness only applies to arguments that are valid.
• A sound argument, in addition to being valid, has all true
premises.
• That is,
Validity + True Premises = Soundness
Some Practice Problems
• Determine whether the following arguments are valid and/or sound
If unicorns are real, then I have
one as a pet.
Unicorns are real.
I have a pet unicorn.
If it’s lower than 15 degrees F out,
water will freeze.
It’s lower than 15 degrees F outside.
Water will freeze.
Badgers are either mammals
or marsupials.
Badgers are mammals.
Badgers are marsupials.
All cars are mammals.
Garfield is a cat.
Garfield is a mammal.
Getting a bit crowded here…
Deductive Arguments
Valid Arguments
So let’s start with a fresh slate:
Deductive Arguments
Valid Arguments
So let’s start with a fresh slate:
Deductive Arguments
Valid Arguments
Where to put the “Sound Arguments” box?
Deductive Arguments
Valid Arguments
Where to put the “Sound Arguments” box?
Deductive Arguments
Valid Arguments
Sound Arguments
Soundness vs Validity Redux
• A valid argument is one where there’s no way the conclusion can
be false if the premises are true.
• A sound argument is a valid argument that has all true premises.
‒Can a valid arguments have a false premise?
‒Can a valid arguments have a false conclusion?
‒Can a sound arguments have a false premise?
‒Can a sound argument have a false conclusion?
Soundness vs Validity Redux
• A valid argument is one where there’s no way the conclusion can
be false if the premises are true.
• A sound argument is a valid argument that has all true premises.
‒Can a valid arguments have a false premise? Yes!
‒Can a valid arguments have a false conclusion? Yes!
‒Can a sound arguments have a false premise? No!
‒Can a sound argument have a false conclusion? No!
• You shouldn’t have to memorize the answers to these questions.
If you know the definitions of validity and soundness, then you
can use them to answer questions like these.
The Division of Labor
• What logicians are interested in is validity, not soundness.
(Remember Aristotle’s big observation!)
• So, to evaluate whether an argument is good, first you have to
decide whether it’s valid. And second, you’ll have to decide
whether the premises are true.
• In this class, we will learn to do the first thing.
• We look to experts (scientists, sports fans, other miscellaneous
aficionados) to do the fact checking (the second thing).
The Plan
• Introduction to the course
‒Logistics ✓
‒Some basics ✓
• The Goal ✓
• The tools we’ll use to reach the goal
‒Possible Worlds
‒Vocab terms
Possible Worlds
• Possible worlds are entities that philosophers have invented to
help them do logic. Things happen in possible worlds.
• For every possible combination of things that happen, there is a
possible world in which that combination of things does happen!
• Possible worlds are maximally specific
‒So, there are a LOT of possible worlds
‒In fact, there are an infinite number of possible worlds
• Philosophers talk about possible worlds A LOT. Some even think
that these alternate possible worlds actually exist.
The Space of all Possible Worlds
• The collection of all Possible Worlds is called “the space of all
possible worlds”
The Space of all Possible Worlds
Trent is tall.
Susie is smart.
The Space of all Possible Worlds
Trent is tall.
Susie is smart.
Trent is not tall.
Susie is smart.
Trent is tall.
Susie is not smart.
Trent is not tall.
Susie is not smart.
The Space of all Possible Worlds
• The collection of all Possible Worlds is called “the space of all
possible worlds”
• We can group Possible Worlds by the features they have in
common.
The Space of all Possible Worlds
Worlds in which Trent is tall.
Worlds in
which
Susie is
not smart.
The Space of all Possible Worlds
• The collection of all Possible Worlds is called “the space of all possible
worlds”
• We can group Possible Worlds by the features they have in common.
• One of the possible worlds is special. That is, our world, or the ‘actual
world’. In the actual world:
‒ The sky is blue.
‒ Zebras have black and white stripes.
‒ Obama wins the 2012 presidential election.
‒ Etc. (remember, PWs are maximally specified, so every fact that is true in
our world is true in the actual world)
• We usually label the actual world ‘α’
The Space of all Possible Worlds
α
Who gets to decide what’s possible?
• The short answer: Logicians.
• The long answer…
Who gets to decide what’s possible?
Logically Possible Worlds:
Laws of Logic
Laws of Mathematics
Definitional Laws
Nomologically Possible Worlds:
Laws of Physics
Actual world
(α)
What is the point?
• We can use PWs to evaluate the validity of an argument.
‒If there is no possible world where the premises are true and
the conclusion false, the argument is valid.
‒If there IS a possible world where the premises are true and
the conclusion false, the argument is invalid.
• No ambiguity about what’s possible and what’s not.
Validity is weird (part III)
• Now that we can talk about validity in terms of PWs, consider
these arguments:
Watermelons are fruits.
2+2=4
Olav is a married bachelor.
Emi likes whales.
• Is there a PW where the premise is true and the conclusion false?
‒No! Weird.
What is going on?
• Sentences like “2+2=4” are true in EVERY possible world…so they’re never
false.
‒ We call these sentences logical truths.
• Sentences like “Olav is a married bachelor” are false in EVERY possible world
(because by definition, bachelors must be unmarried)
‒ We call these sentences logical falsehoods.
• And, FWIW, sentences like “Emi likes whales” are true in some possible worlds,
and false in other possible worlds.
‒ We call these sentences logical contingencies.
Validity Redux
• For the purposes of this class, here’s THE definition of validity:
‒ An argument is valid if and only if there is no possible world where the
premises are true and the conclusion false.
‒ An argument is invalid if and only if there is a possible world where the
premises are true and the conclusion false.
• Validity is weird in 3 ways:
1. It only applies to deductive arguments: Inductive arguments can be good,
but they can’t be valid!
2. Valid arguments can have false premises: Remember Marvin?
3. Finally, if an argument has premises that are never true or a conclusion
that is always true, it’ll automatically be valid. Remember the whales and
watermelons!
Elementary Logic
Emi Okayasu
Summer 2015
The Plan
• Introductions and Introduction to the course
‒Logistics
‒Some basics
• The Goal
• The tools we’ll use to reach the goal
The Syllabus
• Any questions?
Is this class curved?
Is this class curved?
Bad News
Good News
Is this class curved?
Bad News
• Even better news: No, this class is not curved.
Good News
The Plan
• Introductions and Introduction to the course
‒Logistics ✓
‒Some basics
• The Goal
• The tools we’ll use to reach the goal
Why study logic?
• What is logic anyway?
‒Logic is the study of correct reasoning
‒Logic is the beginning of wisdom, not the
end (Leonard Nimoy, aka Spock)
No, but really…what does a logician do?
• Logicians’ primary goal is to detect valid arguments.
• For example:
My dog is friendly.
So, your dog must be friendly too.
vs.
Either Yoshi or Mario won the race.
Yoshi didn’t win, so Mario won the race.
No, but really…what does a logician do?
• Logicians’ primary goal is to detect valid arguments.
• For example:
My dog is friendly.
So, your dog must be friendly too.
X
vs.
Either Yoshi or Mario won the race.
Yoshi didn’t win, so Mario won the race.
✓
Aristotle
• Aristotle (300s BC) was the first
person to systematically study and
categorize valid arguments.
• Aristotelian logic (more on this soon)
remained the gold standard of logic
for over 2000 years.
A note of caution
• Logic can be really tricky. In order to do logic well, we need to be
really precise.
• Specifically, we’ll be defining terms that you might already be
familiar with in new ways.
‒For example, saying that “Your point is valid” is perfectly
acceptable in normal conversation.
‒In this course, however, we will be defining validity a little
differently, so beware!
Arguments
• “An argument is a group of
statements, one or more of which
(the premises) are claimed to provide
support for, or reasons to believe,
one of the others (the conclusion)”
(from page 1 of your textbook)
Examples of Arguments
It’s not the case that both
Sherlock and Moriarty are alive.
Sherlock is not alive. So, Moriarty
is alive.
“Titanium combines readily with
oxygen, nitrogen, and hydrogen, all
of which have an adverse effect on
its mechanical properties. As a
result, titanium must be processed
in their absence.”
-From p. 14 of your book
Premise-Conclusion Form
• Most of the arguments that you’ll encounter in this class will be
presented like this:
It’s not the case that both
Sherlock and Moriarty are
alive.
Sherlock is alive.
Moriarty is not alive.
Titanium combines readily with
oxygen, nitrogen, and hydrogen.
Oxygen, nitrogen, and hydrogen all
have an adverse effect on titanium’s
mechanical properties.
Titanium must be processed in the
absence of oxygen, nitrogen, and
hydrogen.
Validity
• Validity is a property of arguments
‒ (Why a logician would never say you have a “valid point”!)
• An argument is valid when it is impossible for the conclusion to be
false if the premises are true.
• If it’s possible for the conclusion to be false even though all the
premises are true, we say the argument is invalid.
Applying Validity: test yourself
It’s not the case that both Sherlock and
Moriarty are alive.
Sherlock is alive.
Moriarty is not alive.
My dog is friendly.
Your dog is friendly.
Either Yoshi or Mario wins the race.
Yoshi doesn’t win.
Mario wins the race.
Titanium combines readily with
oxygen, nitrogen, and hydrogen.
Oxygen, nitrogen, and hydrogen all
have an adverse effect on titanium’s
mechanical properties.
Titanium must be processed in the
absence of oxygen, nitrogen, and
hydrogen.
The space of all possible arguments
All Arguments
Valid Arguments
The Weirdness of Validity (part I)
• It seems like our definition of validity (which is supposed to be a
good property of an argument) is broken.
• What’s the problem?
‒Validity means that the premises guarantee the truth of the
conclusion. In other words, if the premises are true, then the
conclusion is 100% certain to be true too.
For example…
The sun rose this morning.
The sun rose yesterday morning.
The sun rose the day before yesterday’s morning.
The sun will rise tomorrow morning.
Some arguments are not aiming to establish the truth of the
conclusion 100%. Some arguments are fine with establishing the
truth of the conclusion to 99% (or some other high percentage)
Inductive vs Deductive Arguments
Inductive Arguments
Deductive Arguments
Valid Arguments
Inductive vs Deductive Arguments
Inductive Arguments
Deductive Arguments
Valid Arguments
Deductive Arguments
• In this course, we are only going to focus what makes a good
deductive argument.
‒If you’re interested in what makes a good inductive argument,
take PHILOS 210
• Validity is a property that only applies to deductive arguments.
• You can assume (unless otherwise instructed) that whatever
argument you come across in this course is a deductive argument.
The Plan
• Introduction to the course
‒Logistics ✓
‒Some basics ✓
• The Goal
• The tools we’ll use to reach the goal
The Goal
• Now that we have some of the basics down, we can be more
specific about our goal:
• The goal of this course is to learn how to recognize what makes a
valid (or invalid) deductive argument, and then be able to prove
that a deductive argument is either valid or invalid.
Aristotle’s Observation
An
argument
is valid by
virtue of its
form.
-Aristotle
The form of an argument?
• Consider this argument again:
Either Mario or Yoshi wins
the race.
Yoshi does not win the race
Mario wins the race
The form of an argument?
• Consider this argument again:
Either Mario or Yoshi wins
the race.
Yoshi does not win the race
Mario wins the race
Either I eat museli or yogurt
for breakfast.
I do not eat yogurt for
breakfast.
I eat museli for breakfast.
The form of an argument
• Both these arguments have the same form:
‒ P1 says that there are two possibilities, and at least one of them must
happen.
‒ P2 says that one of the two options doesn’t happen.
‒ So, the conclusion says that the other option must have happened.
• Since both arguments have the same form, they must either both
be valid or both be invalid.
• One of the things we’ll be learning is how to translate an English
argument into a FORMal language (i.e. a language that captures
the form of the argument)
The form of an argument?
• We can represent the fact that these arguments have the same
form by writing them with symbols:
Either Mario or Yoshi wins the race.
Yoshi does not win the race
Mario wins the race
Either I eat museli or yogurt for breakfast.
I do not eat yogurt for breakfast.
I eat museli for breakfast.
MvY
~Y
M
Some Practice Problems
• Determine which of the following are valid. Do any have the same
form?
Pikachu is an electric pokemon.
Pikachu is a pokemon.
Superman can fly.
Superman is Clark Kent.
Clark Kent can fly.
Charizard is a pokemon.
Charizard is a fire pokemon.
All men are mortal.
Socrates is mortal.
Socrates is a man.
Valid vs Invalid Arguments
Deductive Arguments
Valid Arguments
The weirdness of Validity (part II)
• Hopefully you’re getting the hang of validity.
• Now consider this argument:
If there is green cheese on the moon, then Martians will harvest it.
There is green cheese on the moon.
Martians will harvest the green cheese on the moon
• Hopefully I don’t have to convince you that this argument is
completely crazy. BUT, it is valid (!?!)
Why?
Deductive Arguments
Valid Arguments
The weirdness of Validity (part II)
• According to our definition of validity, some pretty crazy
arguments are considered valid.
• But, we don’t want to call the green-cheese/Martian argument a
good argument.
• Validity does not guarantee that an argument is good.
• To help solve this problem, we’re going to introduce another
criteria for evaluating arguments: soundness
Soundness
• Soundness only applies to arguments that are valid.
• A sound argument, in addition to being valid, has all true
premises.
• That is,
Validity + True Premises = Soundness
Some Practice Problems
• Determine whether the following arguments are valid and/or sound
If unicorns are real, then I have
one as a pet.
Unicorns are real.
I have a pet unicorn.
If it’s lower than 15 degrees F out,
water will freeze.
It’s lower than 15 degrees F outside.
Water will freeze.
Badgers are either mammals
or marsupials.
Badgers are mammals.
Badgers are marsupials.
All cars are mammals.
Garfield is a cat.
Garfield is a mammal.
Getting a bit crowded here…
Deductive Arguments
Valid Arguments
So let’s start with a fresh slate:
Deductive Arguments
Valid Arguments
So let’s start with a fresh slate:
Deductive Arguments
Valid Arguments
Where to put the “Sound Arguments” box?
Deductive Arguments
Valid Arguments
Where to put the “Sound Arguments” box?
Deductive Arguments
Valid Arguments
Sound Arguments
Soundness vs Validity Redux
• A valid argument is one where there’s no way the conclusion can
be false if the premises are true.
• A sound argument is a valid argument that has all true premises.
‒Can a valid arguments have a false premise?
‒Can a valid arguments have a false conclusion?
‒Can a sound arguments have a false premise?
‒Can a sound argument have a false conclusion?
Soundness vs Validity Redux
• A valid argument is one where there’s no way the conclusion can
be false if the premises are true.
• A sound argument is a valid argument that has all true premises.
‒Can a valid arguments have a false premise? Yes!
‒Can a valid arguments have a false conclusion? Yes!
‒Can a sound arguments have a false premise? No!
‒Can a sound argument have a false conclusion? No!
• You shouldn’t have to memorize the answers to these questions.
If you know the definitions of validity and soundness, then you
can use them to answer questions like these.
The Division of Labor
• What logicians are interested in is validity, not soundness.
(Remember Aristotle’s big observation!)
• So, to evaluate whether an argument is good, first you have to
decide whether it’s valid. And second, you’ll have to decide
whether the premises are true.
• In this class, we will learn to do the first thing.
• We look to experts (scientists, sports fans, other miscellaneous
aficionados) to do the fact checking (the second thing).
The Plan
• Introduction to the course
‒Logistics ✓
‒Some basics ✓
• The Goal ✓
• The tools we’ll use to reach the goal
‒Possible Worlds
‒Vocab terms
Possible Worlds
• Possible worlds are entities that philosophers have invented to
help them do logic. Things happen in possible worlds.
• For every possible combination of things that happen, there is a
possible world in which that combination of things does happen!
• Possible worlds are maximally specific
‒So, there are a LOT of possible worlds
‒In fact, there are an infinite number of possible worlds
• Philosophers talk about possible worlds A LOT. Some even think
that these alternate possible worlds actually exist.
The Space of all Possible Worlds
• The collection of all Possible Worlds is called “the space of all
possible worlds”
The Space of all Possible Worlds
Trent is tall.
Susie is smart.
The Space of all Possible Worlds
Trent is tall.
Susie is smart.
Trent is not tall.
Susie is smart.
Trent is tall.
Susie is not smart.
Trent is not tall.
Susie is not smart.
The Space of all Possible Worlds
• The collection of all Possible Worlds is called “the space of all
possible worlds”
• We can group Possible Worlds by the features they have in
common.
The Space of all Possible Worlds
Worlds in which Trent is tall.
Worlds in
which
Susie is
not smart.
The Space of all Possible Worlds
• The collection of all Possible Worlds is called “the space of all possible
worlds”
• We can group Possible Worlds by the features they have in common.
• One of the possible worlds is special. That is, our world, or the ‘actual
world’. In the actual world:
‒ The sky is blue.
‒ Zebras have black and white stripes.
‒ Obama wins the 2012 presidential election.
‒ Etc. (remember, PWs are maximally specified, so every fact that is true in
our world is true in the actual world)
• We usually label the actual world ‘α’
The Space of all Possible Worlds
α
Who gets to decide what’s possible?
• The short answer: Logicians.
• The long answer…
Who gets to decide what’s possible?
Logically Possible Worlds:
Laws of Logic
Laws of Mathematics
Definitional Laws
Nomologically Possible Worlds:
Laws of Physics
Actual world
(α)
What is the point?
• We can use PWs to evaluate the validity of an argument.
‒If there is no possible world where the premises are true and
the conclusion false, the argument is valid.
‒If there IS a possible world where the premises are true and
the conclusion false, the argument is invalid.
• No ambiguity about what’s possible and what’s not.
Validity is weird (part III)
• Now that we can talk about validity in terms of PWs, consider
these arguments:
Watermelons are fruits.
2+2=4
Olav is a married bachelor.
Emi likes whales.
• Is there a PW where the premise is true and the conclusion false?
‒No! Weird.
What is going on?
• Sentences like “2+2=4” are true in EVERY possible world…so they’re never
false.
‒ We call these sentences logical truths.
• Sentences like “Olav is a married bachelor” are false in EVERY possible world
(because by definition, bachelors must be unmarried)
‒ We call these sentences logical falsehoods.
• And, FWIW, sentences like “Emi likes whales” are true in some possible worlds,
and false in other possible worlds.
‒ We call these sentences logical contingencies.
Validity Redux
• For the purposes of this class, here’s THE definition of validity:
‒ An argument is valid if and only if there is no possible world where the
premises are true and the conclusion false.
‒ An argument is invalid if and only if there is a possible world where the
premises are true and the conclusion false.
• Validity is weird in 3 ways:
1. It only applies to deductive arguments: Inductive arguments can be good,
but they can’t be valid!
2. Valid arguments can have false premises: Remember Marvin?
3. Finally, if an argument has premises that are never true or a conclusion
that is always true, it’ll automatically be valid. Remember the whales and
watermelons!
